QuestionUse properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1 . Where possible, evaluate logarithmic expressions. $\begin{array}{c} \log x+\log \left(x^{2}-4\right)-\log 5-\log (x+2) \\ \log x+\log \left(x^{2}-4\right)-\log 5-\log (x+2)= \end{array}$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. The expression involves logarithms that can be combined using logarithmic properties.
2. The properties of logarithms such as the product rule, quotient rule, and power rule will be useful.
3. The expression should be simplified to a single logarithm with a coefficient of 1.

STEP 2

1. Apply the product rule of logarithms to combine terms.
2. Apply the quotient rule of logarithms to condense the expression.
3. Simplify the expression further if possible.

STEP 3

Apply the product rule of logarithms, which states that $\log a + \log b = \log (ab)$ , to the first two terms:
$\log x + \log (x^2 - 4) = \log (x(x^2 - 4))$
This simplifies to:
$\log (x^3 - 4x)$

STEP 4

Apply the quotient rule of logarithms, which states that $\log a - \log b = \log \left(\frac{a}{b}\right)$ , to the entire expression:
$\log (x^3 - 4x) - \log 5 - \log (x+2) = \log \left(\frac{x^3 - 4x}{5(x+2)}\right)$

STEP 5

Check if further simplification is possible. In this case, the expression is already simplified as a single logarithm:
$\log \left(\frac{x^3 - 4x}{5(x+2)}\right)$
Since there are no further simplifications or evaluations possible, the expression is fully condensed.
The final condensed expression is:
$\boxed{\log \left(\frac{x^3 - 4x}{5(x+2)}\right)}$

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